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International Journal of Impact Engineering 35 (2008) 1293–1302 www.elsevier.com/locate/ijimpeng
On the application of genetic algorithms for optimising composites against impact loading M. Yong, B.G. Falzon, L. Iannucci Department of Aeronautics, Imperial College, London SW7 2AZ, UK Received 27 December 2006; received in revised form 8 October 2007; accepted 9 October 2007 Available online 17 October 2007
Abstract A genetic algorithm (GA) was adopted to optimise the response of a composite laminate subject to impact. Two different impact scenarios are presented: low-velocity impact of a slender laminated strip and high-velocity impact of a rectangular plate by a spherical impactor. In these cases, the GA’s objective was to, respectively, minimise the peak deflection and minimise penetration by varying the ply angles. The GA was coupled to a commercial finite-element (FE) package LS DYNA to perform the impact analyses. A comparison with a commercial optimisation package, LS OPT, was also made. The results showed that the GA was a robust, capable optimisation tool that produced near optimal designs, and performed well with respect to LS OPT for the more complex high-velocity impact scenario tested. r 2007 Dr Matthew Yong. Published by Elsevier Ltd. All rights reserved. Keywords: Genetic algorithms; Impact; Composites; Optimisation; Finite elements
1. Introduction The increasing use of composites in high-performance and weight-critical applications has seen a corresponding increase in the number of impact events involving these materials. Unfortunately, even minor impact events can cause a significant reduction particularly in the residual compressive strength [1], and this is a cause for concern where structural integrity is critical. Alternatively, the configurable, lightweight properties of composite materials make them attractive choices in designs subject to impact threats. The main goals of optimising a structure against impact are to improve damage resistance, damage tolerance, or energy absorption. Damage resistance is a property that enables a structure to resist the onset of damage. This may be of interest in structures where impact events do not feature in their primary role, such as skin panels on aircraft. Damage tolerance enables a structure to resist the effects of damage. This is of interest in structures where being hit is likely and the subsequent post-impact Corresponding author. Tel.: +44 20 7594 5100; fax: +44 20 7584 8120.
E-mail address:
[email protected] (M. Yong).
performance is important. A possible use of damagetolerant designs would be in dual-purpose armour panels for light vehicles that also serve some structural role. Examples of energy-absorbing designs would be helicopter subfloors and automotive crash barriers, where the goal is to dissipate as much of the kinetic energy as possible.
2. Optimisation studies Despite the potential weight, cost and performance advantages of optimisation against impact, limited work has been done on optimising fibre-reinforced, polymer matrix composite structures against impact threats. In the wider context of impact optimisation, most of the improvements tend to be experimentally found, which involves testing a limited sample of the vast design space [2,3]. The conclusions are thus locally optimal and specific to the test circumstances. Other approaches use analytical techniques to determine the optimal design [4,5], with the associated limitations of theory on model complexity and accuracy. The limited numerical optimisation work on composite armour may be partly due to the numerical cost associated
0734-743X/$ - see front matter r 2007 Dr Matthew Yong. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2007.10.004
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Nomenclature eab nab y sab tab
strain in the ab-plane Major Poisson’s ratio off-axis angle stress in the ab-plane shear stress in the ab-plane
with even one impact simulation, as suggested by [6,7]; explicit schemes would necessarily require a fine mesh discretisation and short time steps to capture the rapidly varying stresses in space and time. It will be shown later that optimal designs often resist projectile penetration via gross deformation over extended times—meaning that impact simulations for optimal designs may be significantly longer than those for penetrated designs. An optimisation process involving hundreds, if not thousands, of numerical simulations would quickly become intractable. Furthermore, there is a lack of fully developed, robust models capable of capturing the complex physics of composites undergoing impact that results in progressive damage and ultimate failure. Nevertheless, as material models improve and computing power increases, optimisation via numerical simulations is attractive because of the reduced cost per solution and a much larger variety of designs that can be considered. The tests on the tools presented here are done in expectation of such improvements. Genetic algorithms (GAs) are search techniques based on natural selection and population genetics. Selection, crossover and occasionally mutation are applied to a population of candidate solutions sampled from the larger design space, with fitter individuals having a higher probability of reproducing. Genes that predispose certain individuals to have improved fitness are thus sampled more frequently and propagated throughout the population. GAs are robust optimisers that can handle real/discrete variables. Detailed treatments of this ever-growing field may be found in textbooks [8–10]. GAs are also easily parallelised to a variety of useful parallel topologies, thus reducing computation times or even improving solution quality at equivalent cost [11]. A GA, coupled with response surface approximations, has been used to minimise a ballistic shield’s mass subject to penetration energy and displacement constraints [6]. Approximations between an initial sampling of exact finiteelement (FE) solutions were used to cut down the computational costs of the GA’s trial solutions, with further exact FE solutions being subject to convergence between the approximated and FE-evaluated solutions. Significant non-linearity even to small changes of the design variables and the potential of GAs for impact optimisation due to their robust global search was noted by the authors here. A GA was recently used to optimise a laminate against low-velocity (p12 m s1) impact, by minimising delamination and matrix cracking damage
Gab Sc Xc Xt Yc Yt
shear modulus in the ab-plane shear strength longitudinal compressive strength longitudinal tensile strength transverse compressive strength transverse tensile strength
subject to weight and cost considerations [12]. An in-house FE code was used to analyse the designs. An approximate optimisation or response surface method in ANSYS coupled with LS DYNA was also used to vary the projectile impact angle and laminate thicknesses of ceramic/steel/ aluminium armour for minimum penetration [7]. Optimising a composite laminate against an impact threat is in general a multiobjective problem with multiple real and discrete variables. The real variables may be structural dimensions, ply thicknesses and angles, while the integer variables may be ply materials. Manufacturing practicalities may convert otherwise real variables into discrete ones—such as ply angles only being allowed to vary in 301 increments or an integer multiple of plies with specified thickness. For example, a laminate’s stacking sequence was optimised subject to strength/stiffness and weight considerations [13]. As a laminate’s bulk properties depends on the angle of each ply, the search space is also littered with local optima, as graphically seen in the cited work with a simple four-ply symmetric laminate. As such, a search technique that is robust and can handle many integer/real variables is required. 3. Impact models 3.1. Overview The first optimisation scenario involved the low-velocity transverse impact of a cylindrical impactor on a 125 10 1 mm, eight-ply symmetric laminate strip modelled with thin shell elements, with an impactor velocity of 5 m s1 and the strip being clamped at both ends. Shell elements were used to model the strip as its thickness was small and the behaviour is expected to be dominated by inplane stresses. The impactor in both cases was modelled using rigid solid elements. Tests with different mesh densities showed minor differences between a 125 13 and 50 1 mesh. As such, a 50 1 mesh was used as the optimisation model. The low-velocity model is illustrated in Fig. 1. Material properties are described in Table 1, while the mesh parameters are presented in Table 3. The second optimisation scenario was the impact of a rectangular 100 75 2 mm laminate by a rigid 6 mm diameter spherical impactor moving at 225 m s1; see Fig. 2. The material properties are described in Tables 1 and 2, and the mesh parameters are described in Table 4. The laminate was clamped on all four sides. The
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angles in a 16-ply symmetric laminate. In both cases, ply angles were varied in 11 increments to challenge the optimisation algorithms with a vast search space. However, the GA is capable of handling a coarser degree of discretisation, which may be necessary due to manufacturing limitations. 3.2. Composite material model
Fig. 1. Low-velocity impact model [33].
Table 1 Composite plate material properties (typical CFRP) Density Longitudinal Young’s modulus, Exx Transverse Young’s modulus, Eyy Poisson’s ratio, nxy Shear modulus, Gxy Longitudinal compressive strength, xc Longitudinal tensile strength, xt Transverse compressive strength, yc Transverse tensile strength, yt In plane shear strength, sc
1500 kg m3 119 GPa 9.2 GPa 0.28 4.6 GPa 1350 MPa 2005 MPa 194 MPa 69 MPa 80 MPa
In order to improve the validity of the solutions, unidirectional (UD) carbon/epoxy composite plies were modelled using a non-interactive, progressive composite damage model (Material #54) in LS DYNA. The Chang–Chang failure criteria are used in this material model, which can represent tensile/compressive fibre and matrix failure. When a given failure criteria for a given failure mode is met, appropriate stiffnesses and Poisson ratios are set to zero. Referring to [14], the criteria for tensile fibre failure is given by 2 s11 s12 2 þb ef ¼ 1 (1) Xt Sc If the condition that the local stress is tensile in the fibre direction and e2f X 0 is met, then E11 ¼ E22 ¼ G12 ¼ n21 ¼ n12 ¼ 0. For compressive fibre failure, 2 s11 2 ec ¼ 1 (2) Xc If the element stress is compressive in the fibre direction and e2c X0, then E11 ¼ n21 ¼ n12 ¼ 0. For tensile matrix failure, 2 2 s22 s12 e2m ¼ þ 1 (3) Yt Sc If the element stress is tensile in the transverse direction and e2mX0, then E22 ¼ n21 ¼ G12 ¼ 0. Finally, for compressive matrix failure, # 2 " 2 2 s22 Yc s22 s12 2 ed ¼ þ 1 þ 1 (4) 2S c 2S c Yc Sc
Fig. 2. High-velocity impact model [33].
Table 2 Steel impactor material constants Density Stiffness Poisson’s ratio
7850 kg m3 207 GPa 0.3
optimisation process was to minimise penetration velocity in the case of penetration, or maximise rebound velocity in the case of rebound. Maximising the rebound velocity would minimise the amount of energy absorbed by the plate. This was to be accomplished by varying eight ply
For a compressive transverse element stress, if e2dX0, then E22 ¼ n21 ¼ n12 ¼ G12 ¼ 0 and Xc ¼ 2Yc. Degradation of the mechanical properties occurs over a finite number of time steps. The degradation of these values over a fixed number of time steps causes mesh sensitivity, as smaller elements with shorter time steps degrade in an apparently more brittle manner than larger elements with longer time steps [15]. In order to proceed, a further two parameters, a and b, of the composite model used had to be ascertained. a governs the degree of non-linearity of the shear stress– strain relationship. An otherwise linear shear stress–strain relationship indicated in Fig. 4 exists with a ¼ 0. A suitable value of a was found by calibrating the polynomial shear
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8-Node solid element Rigid (Mat #20) 4-Node membrane/shell element Belytschko–Tsay Enhanced composite damage (Mat #54) 1920 2453 50 101 8 (one per ply) 8 7050 ms Automatic surface to surface 67:1
Table 4 High-velocity case—FE parameters Element type for impactor Material type Solid formulation Element type for plate Shell formulation Material type Number of solid elements Number of solid element nodes Number of shell elements Number of shell element nodes Number of integration points Total number of plies Simulation end time Contact logic Impactor:plate mass ratio
8-Node solid element Rigid (Mat #20) 1-Point corotational 4-Node membrane/shell element Belytschko–Tsay Enhanced composite damage (Mat #54) 2048 2663 1200 1270 16 (one per ply) 16 500 ms Automatic surface to surface 0.039:1
Fig. 3. Mesh convergence of second model.
stress–strain relationship in Eq. (5), used in the numerical model [16], with experimental shear stress–strain plots found in [17] for the same material system: 212 ¼
t12 þ at312 G 12
(5)
An iterative error-minimising approach was used to estimate a value of 1.58 108 (g cm1 ms2)3 for a. From Eq. (1), we see that parameter b in this material model governs the significance of shear strength in fibre tensile failure. In order to estimate a suitable value of b, which is defined to be between zero and unity, a oneelement model was loaded in off-axis tension to generate failure stresses for different b over a range of angles varying from 01 to 451. The equation for off-axis failure strength from [18] is shown below: " ! #ð1=2Þ cos4 y 1 1 sin4 y 2 2 sy ¼ sin y cos y þ þ (6) X 2t Y 2t S 2c X 2t Fig. 5 plots the failure strength against off-axis angle. For clarity, other values of b ranging between 0.0 and 1.0 were omitted; however, the trend is that the numerical results converge to the theoretical ones with increasing b. From these tests, it was found that the best value of b to use was about 1.0.
3.3. Justification for shell elements If meshed methods are to be used for modelling impact, solid rather than shell elements are preferred, as solid elements allow a more accurate representation of throughthickness effects and damage. For relatively light projectiles at velocities well above the ballistic limit, the response will be dominated by local effects. The influence of the structural support and global properties increases as heavier impactors are used at velocities close to or below the ballistic limit. Nevertheless, as this is a parametric and comparative study with otherwise identical runs being repeated with different random seeds multiple times, keeping the total computational costs down necessitated the use of cheaper shell elements. The tools used remain equally valid if solid rather than shell elements are used. Therefore, thin Belytschko–Tsay shell elements were also used to model the gross behaviour of the plate. Although such elements would not be able to resolve transverse shear strains and through-thickness stresses, it proved necessary to use these elements to keep computation times manageable. It is thus expected that the low-velocity impact FE model with a large impactor:plate mass ratio, much longer contact times and corresponding gross, global deformations would be more representative of reality than the highvelocity model with a small impactor:plate mass ratio, shorter contact times and more localised behaviour [19]. Fig. 3 shows the mesh convergence of the second model. As this is a comparative/parametric study, a 40 30 mesh (71 s) is used in preference to the more accurate 60 40 one (502 s), thus reducing computation times by roughly a factor of 7. The 80 60 mesh took 1516 s while the 100 75 mesh took 1904 s.
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Plot of shear stress against shear strain 80 70
Shear stress, τ xy /MPa
60 50
Experimental FE Best cubic fit
40
FE Linear
30 20 10 0 0
0.01
0.02 0.03 Shear strain, ε xy
0.04
0.05
Fig. 4. Experimental/model shear stress–strain plot.
Plot of failure stress against angle 2500.00
Failure strength or σθ /MPa
2000.00
1500.00
Theoretical
1000.00
beta = 0 beta = 1.0 500.00
0.00 0
5
10
15
20
25
30
35
40
45
Angle θ /degrees Fig. 5. Theory/FE off-axis failure envelope.
4. Procedure LS DYNA [20,21] was coupled with an in-house GA [22] written in Fortran, in order to evaluate and optimise the designs. The results were then compared with LS OPT [23], an optimisation package also available from LSTC. A flowchart illustrating the key stages of the optimisation procedure with the GA is shown in Fig. 6. A high-quality pseudorandom number generator (PRNG) with a sufficiently large period must exist at the core of the GA and indeed in most stochastic optimisation techniques, so that artificial bias due to recurrent patterns
in the random numbers is minimised. LS OPT uses a PRNG based on the Mersenne Twister Algorithm [24]. The GA was tested using both the Mersenne Twister Algorithm [25] having a period of 219937-1 and a simple PRNG based on the Park and Miller technique with a Bays–Durham shuffle with a period of 2.3 1018 [26]. The GA used in the following impact optimisation work had a generational replacement strategy with a standard binary representation scheme. It begins with an initial population that consists of a randomised gene pool. Each individual genome consists of genes and represents a potential solution to the problem. Alternatively, users with
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velocity test was chosen so that the global optimum was known—an all 01 stacking sequence. In this benchmark the genes represented the stacking sequence of the laminate, with crossover effectively acting to transplant beneficial ply combinations between designs. Three different sampling techniques were tested with LS OPT: polynomial D-optimal, Monte Carlo sampling and neural network sampling. For a symmetric laminate the search space was approximately 1.05 109. The results are presented after tests using six distinct starting seeds for the pseudorandom number generator. In this problem, LS OPT (Tables 5–7) performed better than the GA (Tables 8–10). The maximum displacements
Fig. 6. GA optimisation flowchart.
problem-specific knowledge may decide to seed the starting population with a variety of good known solutions [27]. Among the population, fitter individuals are preferentially selected as parents for mating. Occasionally, a small mutation rate was applied, resulting in mutant designs. Mutation helps to prevent genes becoming irreversibly lost and also performs a form of random hill climbing in the vicinity of the solutions found. Various population sizes were tested and the algorithm was stopped after a predetermined maximum generation. Possible solutions were checked against a database of past analyses to avoid repeating FE simulations should a match be identified, as done by other authors [28]. If a candidate solution was unique, an FE analysis was performed. As the population sizes were small, the computational overhead of doing such a check was outweighed by the expensive cost of a FE evaluation. Checking for repeat solutions yielded a considerable reduction in the number of FE evaluations required per optimisation run, due to the population becoming increasingly homogeneous as it converged. It also allowed the user to simply set the initially unknown ideal end generation to a suitably large figure—once genetic uniformity had occurred, barring mutants, all subsequent individuals would be excluded from a costly FE analysis. Consequently, as an optimisation technique that makes relatively few assumptions about the nature of the design space, the only key remaining parameters would be the population size and mutation rates, while other lower-level parameters such as gene lengths, discretisation levels, and the encoding of introns may be left at suggested values.
5. Results and discussion 5.1. Low-velocity impact case The goal of the optimisation process was to minimise the peak deflection during impact by allowing LS OPT or the GA to alter the layup. The configuration of the low-
Table 5 Low velocity case—polynomial D-optimal results FE evals.
Best layup
Max. disp.
21 31 41
[4, 4, 90, 4]s [90, 30, 30, 90]s [0, 90, 90, 90]s
0.63027 0.78785 0.77153
Table 6 Low velocity case—Monte Carlo results FE evals.
Best layup
26 101 151 221
[4, 0, 4, [6, 7, 3, [1, 4, 7, [1, 4, 3,
Max. disp. 4]s 3]s 4]s 4]s
0.59747 0.59950 0.59918 0.59628
Table 7 Low velocity case—neural network results FE evals.
Best layup
Max. disp.
26 91 181
[2, 15, 4, 20]s [1, 2, 0, 5]s [1, 0, 3, 1]s
0.61801 0.59636 0.59577
Table 8 Low velocity case—one-point crossover GA results with Park and Miller RNG Genetic algorithm results—Park and Miller RNG Population Mutation (%) FE evals.
Best disp. Avg. disp.
10 10 10 15 15 15 20 25
0.64044 0.62900 0.59901 0.65461 0.62404 0.59468 0.59619 0.61690
0.0 0.1 1.0 0.0 0.1 1.0 0.0 0.0
[28.5, 6.25] [36.7, 8.36] [78.5, 8.46] [46.7, 6.53] [64.5, 9.42] [165, 10.6] [66.0, 15.5] [84.2, 14.0]
[0.66748, [0.64981, [0.62204, [0.67197, [0.64776, [0.59879, [0.63337, [0.62484,
Numbers in parenthesis denote [average, standard deviation].
0.0240] 0.0157] 0.0179] 0.0195] 0.0202] 0.0045] 0.0247] 0.0187]
ARTICLE IN PRESS M. Yong et al. / International Journal of Impact Engineering 35 (2008) 1293–1302 Table 9 Low-velocity case—one-point crossover GA results with Mersenne Twister RNG Genetic algorithm results—Mersenne Twister RNG Best disp. Displacements
10 10 10 15 15 15 20 25
0.64242 0.63835 0.60054 0.63571 0.61944 0.59469 0.61075 0.60211
[27.7, 5.99] [34.0, 5.32] [68.5, 6.02] [52.2, 11.5] [60.3, 8.94] [159, 18.1] [82.8, 25.7] [98.3, 13.0]
significant sample size, but based on these six sets of tests, the two RNGs appear roughly comparable. 5.2. High-velocity impact case
Population Mutation (%) FE evals. 0.0 0.1 1.0 0.0 0.1 1.0 0.0 0.0
1299
[0.66842, [0.66436, [0.62175, [0.65316, [0.63954, [0.60442, [0.64079, [0.62031,
0.0233] 0.0191] 0.0155] 0.0202] 0.0198] 0.0133] 0.0229] 0.0109]
Table 10 Low-velocity case—two-point crossover GA results with Park and Miller RNG Genetic algorithm results—Park and Miller RNG Population size
Mutation rate (%)
FE evals.
Max. disp.
10 10 10 15 15 15 20 25
0.0 0.1 1.0 0.0 0.1 1.0 0.0 0.0
17 27 64 33 48 141 38 83
0.67731 0.67731 0.64357 0.67731 0.67731 0.65642 0.67731 0.66250
of the Monte Carlo and neural network techniques in particular are good, with layups close to the ideal. This could be due to the independent nature of the ply angles and the monotonic variation of global stiffness with any ply angle. Maximum stiffness of the laminate to out-ofplane bending can be sequentially achieved by reducing the ply orientations of any ply, in any order, to 01. Of the three sampling techniques tested, the Monte Carlo method appeared best for smaller test sets while the neural network had better results if it was permitted to sample larger sets. It was empirically observed that one-point crossover produced better results than two-point crossover; cf. Tables 8 and 9 with Table 10. This is contrary to recommendations by various authors [29]. In laminate stacking sequence problems where the outer plies have a bigger influence on out-of-plane behaviour compared with the central plies, it may be prudent to emphasise increased genetic exploration of the outermost plies. If genes at the start of the genome map to ply angles at the outermost portion of the laminate, then one-point crossover with its first crossover point anchored at the start of the genome will naturally emphasise genetic exchange of the outer plies. The six tests with different random starting seeds are presented in Tables 8 and 9, using the Park and Miller and Mersenne Twister RNGs, respectively. Additional computational resources are necessary for a more statistically
In the high-velocity impact scenario, the ply angles of the laminate were varied in order to minimise the exit velocity should penetration occur, or maximise the rebound velocity should rebound occur. As before, the stacking sequence was represented as a sequence of genes. In this test case, the ideal stacking sequence was not known and the relevant measure of merit of the optimisation technique is their relative performance. For a 16-ply symmetric laminate, the search space was suitably large at approximately (180)8 or 1.10 1018. It is worth noting that the conventional crossply layup had penetration at 103 m s1 and the quasi-isotropic layup had penetration at 20.0 m s1, respectively [21]. Tables 11–13 display the results obtained using three LS OPT techniques. The three LS OPT methods consistently achieved similar values and appear quite comparable. Table 14 gives the results of the GA optimisation process. Based on these results, the GA appears to compare well with LS OPT. The population 10, mutation 1.0% results with 100 FE evaluations had an average rebound velocity slightly higher than the experiment 10, iteration 10 LS OPT results. Increasing the number of FE evaluations did not seem to make a significant difference for LS OPT’s results, while the GA results obtained with 223 FE evaluations using a population size of 15 and a mutation rate of 1.0% Table 11 High-velocity case—LS OPT results, polynomial sampling
Expt. 10, Iter. 10 Expt. 15, Iter. 15 Expt. 20, Iter. 20
FE evals.
Rebound velocity (m s1)
[84, 32] [187, 78] [207, 131]
[56.3, 13.2] [56.3, 13.2] [56.9, 13.0]
Table 12 High-velocity case—LS OPT results, Monte Carlo sampling
Expt. 10, Iter. 10 Expt. 15, Iter. 15 Expt. 20, Iter. 20
FE evals.
Rebound velocity (ms1)
[101, 1] [226, 1] [394, 17]
[55.6, 5.32] [55.6, 5.32] [55.7, 5.11]
Table 13 High-velocity case—LS OPT results, neural network sampling
Expt. 10, Iter. 10 Expt. 15, Iter. 15 Expt. 20, Iter. 20
FE evals.
Rebound velocity (m s1)
[101, 0] [226, 0] [379, 34]
[52.6, 1.57] [54.0, 1.84] [54.9, 2.06]
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produced the best set of consistently good results at 64.7 m s1. An illustration of the crossply and GA optimised is contrasted in Fig. 7. In general it was observed that mediocre performance occurred with mutation-free populations, while the best results for a given population size consistently occurred at
Table 14 High-velocity case—GA results Population size
Mutation rate (%)
FE evals.
Rebound velocity (m s1)
10 10 10 15 15 15 20 20 25
0.0 0.1 1.0 0.0 0.1 1.0 0.0 0.1 0.0
[44, 8] [58, 10] [100, 26] [81, 29] [124, 29] [223, 27] [118, 48] [182, 56] [164, 37]
[44.1, [49.5, [57.2, [45.1, [51.5, [64.7, [54.4, [56.3, [56.0,
4.69] 7.24] 9.09] 3.18] 5.43] 6.70] 4.19] 5.82] 2.53]
increasing mutation rates. Of the three mutation probabilities tested, the highest rate at 1% produced the best results, and is consistent with suggestions by other authors [10]. Unfortunately, due to computational cost constraints, more comprehensive tests on population size and mutation rate could not be performed. In Fig. 8, the velocity histories of two optimal layups are compared with a crossply benchmark. The first GAoptimised layup has a similar quality (55 m s1) to that found by LS OPT. The second has an improved rebound velocity of 62.6 m s1. The GA found these solutions in 50 and 70 FE evaluations, respectively, compared with LS OPT’s 84–394 FE evaluations. Note that the crossply laminate produced a short, sharp impact event, while an optimal layup produced a much more protracted impact event. Table 15 shows the membrane and flexural stiffness of the different laminates, calculated using the Laminate Analysis Program [30]. The latter two GA optimised layups for this test case indeed appear to be more compliant than the former two baseline crossply and quasi-isotropic layups. Based on these results, if we seek to improve the
Fig. 7. Lower surface of crossply laminate and upper surface of GA-optimised plates, respectively [33]. Impactor penetrates crossply laminate at 110 m s1 while rebound begins on optimised plate later at 200 ms.
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Velocity history of optimised and crossply layups 100 50
Velocity /ms-1
0 0
50
100
150
200
250
300
350
400
450
500
-50 -100 -150 -200 -250 Time /us
Fig. 8. Impactor velocity histories for the GA-optimised and crossply layups. Note the positive and negative exit velocities, respectively.
Table 15 Plate stiffness values [30] Laminate details
Baseline: Baseline: Optimal: Optimal:
crossply quasi-isotropic 50 FE evals., 55 m s1 rebound [68,101,21,56,160,35,8,147]s 96 FE evals., 66.8 m s1 rebound [61,9,36,23,153,81,92,53]s
ballistic limit, it appears desirable to produce a compliant structure that experiences a protracted impact event with lower stresses and larger deformations, than a stiff structure that undergoes a sharp impact event with higher stresses and consequent damage initiation.
6. Conclusions Based on the work presented, the GA is a robust optimiser that successfully located optimal solutions and appeared to perform best with small levels of mutation along with one-point crossover, which focused genetic exchange on the significant outermost plies. The GA can essentially be used for challenging problems with limited problem/domain-specific knowledge and the user can expect a reasonable answer. This agrees with recommendations by other authors [31]. The three LS OPT sampling techniques tested appear to be a better option for problems involving variables that are not strongly inter-related, where the optimal solution may be obtained by sequentially altering the constituent variables in an arbitrary order, such as the low-velocity impact test case. In the tests performed, the GA performed better in the high-velocity impact scenario where the search space is larger and non-linear.
Membrane stiffness (1010 N m2)
Flexural stiffness (1010 N m2)
Exx
Eyy
Exx
Eyy
6.42 4.63 5.26 4.27
6.42 4.63 3.66 4.28
5.39 6.04 3.08 4.77
7.45 3.85 5.92 2.61
LS OPT appears capable of quickly finding promising solutions but suffers from a high premium on marginally improving them or gainful exploration; for example, expending roughly 100–300 extra FE evaluations only improved results by 0.1–2.3 m s1, or roughly 3%. This is negligible in contrast to the variance between runs with different starting seeds (up to 23%)—the user may be better off testing other seeds rather than tweaking the parameters affecting the run’s duration! For the GA with a smaller population size of 10, increasing the average FE evaluations from 44 to 100 consistently improved the rebound velocities by 13.1 m s1 or about 30%. Similarly, for a population size of 15, increasing the average FE cost from 81 to 223 improved the rebound velocity by 19.6 m s1 or about 43%. As such, based on these results, we can say that if the user is willing to pay the computational cost, they can consistently obtain better solutions with the GA than LS OPT. While the practical utility of a GA’s optimal designs are limited by the accuracy of the fitness analysis, as it makes few assumptions about the nature of the problem, it remains as capable regardless of the model’s accuracy. Nevertheless, more accurate constitutive models are necessary to improve confidence in the results. Computational costs aside, a solid-element model is preferable for actual optimisation work due to its ability to resolve
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